Assume the square root of 2 is a fraction, and watch the assumption collapse

What to assume they know

• A prime dividing a square must divide the number itself — that a prime dividing a square divides the number
• What a rational number is: a ratio of two integers with a non-zero bottom
• Reducing a fraction to its simplest form, and what it means for two integers to share no factor beyond 1
• Squaring both sides of an equation, and rearranging a two-term equation
• From Class IX: that irrational numbers exist and can be placed on the number line, though no proof of irrationality was given there

What they should be able to do

• State what has to be shown to call a number irrational, and explain why it is a claim about every fraction at once
• Explain why a decimal expansion, however far it is carried, settles nothing
• Set up a proof by contradiction: name the assumption being made for the sake of argument, and keep it visible
• Justify the reduction to a fraction whose two parts share no factor, and say why the proof would collapse without it
• Carry the algebra from the assumed equality to the statement that 2 divides the numerator's square
• Apply Theorem 1.2 at each of the two points where it is needed, naming the prime used
• Identify the contradiction precisely, and state what has been refuted — the assumption, not any of the steps
• Run the same argument on 4 and locate the exact line at which it stops working

Where it usually goes wrong

• "The decimal goes on forever, so it is irrational." One third also goes on forever and is a ratio of two integers. What matters is whether the digits eventually repeat.
• "Assuming what you want to disprove is cheating." It is the technique. You are not claiming the assumption; you are testing it to destruction, and the chapter names the method and points at where it is discussed in more detail.
• "The coprime step is just tidying." It is the load-bearing step. Without it, deriving that both parts are even is not a contradiction at all — plenty of fractions have two even parts. Take the step away and show the proof deflate.
• "2 divides the numerator's square, so obviously 2 divides the numerator." This is the one line that needs a theorem, and it is false for composite divisors. If a student finds it obvious, the counterexample from A prime dividing a square must divide the number itself — 4 dividing 36 but not 6 — is the corrective.
• "The proof shows the algebra was wrong somewhere." Every line after the assumption is valid. That is what forces the blame back onto the assumption, and it is the part students most often garble in the written answer.
• "This proves square roots are irrational." It proves one square root is, and the run on 4 shows why the argument cannot be waved at all of them.

Questions to check understanding

• Prove that the square root of 2 is irrational, with the coprime assumption stated explicitly
• Identify, in a supplied proof, the step at which Theorem 1.2 is used and the prime it is used with
• Explain what the contradiction refutes and why the intermediate steps stand
• Say what would go wrong if the fraction were not first reduced
• Explain why the same argument does not show the square root of 4 is irrational

Examples worth working on the board

Values marked verified are worked out here; the chapter prints no answers, and no answer key was consulted.

• The proof's skeleton (§1.3, pp. 6–7). Inputs, in the order the chapter uses them: assume the square root of 2 equals one integer over a non-zero integer; cancel any shared factor so the two are coprime; multiply up and square, reaching twice the denominator's square equal to the numerator's square; deduce that 2 divides the numerator's square; apply Theorem 1.2 to get that 2 divides the numerator; write the numerator as twice a new integer; substitute; simplify to the denominator's square equal to twice the new integer's square; apply Theorem 1.2 again to get that 2 divides the denominator. The two parts now share 2.
• The substitution in numbers. Verified: if the numerator is 2c then its square is 4c², so twice the denominator's square equals 4c², and dividing by 2 leaves the denominator's square equal to 2c². The factor of 2 has moved from one side to the other, which is exactly why the argument repeats.
• Why decimals cannot decide (not in the book). Verified: 99/70 = 1.414285714…, and its square is 9801/4900 = 2.000204…; 577/408 = 1.4142156…, and its square is 332929/166464 = 2.0000060…. Both are close and neither is 2. No amount of closeness is evidence, and this panel is what makes the proof feel necessary rather than pedantic.
• A second way to see the same obstruction (not in the book). List the squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and the doubled squares 2, 8, 18, 32, 50, 72, 98, 128, 162, 200. Verified: no value appears in both lists, and the equation the proof reaches is precisely a demand that one should.
• The stress test on 4 (not in the book, and the most important input here). Assume the square root of 4 is a coprime fraction. Then four times the denominator's square equals the numerator's square, so 2 divides the numerator's square, so by Theorem 1.2 the numerator is 2c. Substituting gives four times the denominator's square equal to 4c². Verified: dividing by 4 leaves the denominator's square equal to c², not to twice anything — there is no factor of 2 left to hand on, and the argument simply stops. And it must, because the square root of 4 is 2, which is 2 over 1 with nothing shared. Show the two runs side by side and let the divergence appear at one line.
• The other roots this proof unlocks (§1.3, pp. 7–9). Once the square root of 2 is settled, the chapter re-runs the same argument for 3 on pp. 7–8, sets 5 as Exercise 1.2 Q1 on p. 9, and uses the settled result for 2 again in Example 7 on p. 8. That downstream traffic belongs to Re-running the same contradiction on other prime square roots, but it is worth naming here so the proof does not look like a one-off.

Figures to have open

• A parallel-column layout that can run two versions of the same proof line by line and mark the line where they diverge. This is an added device and the topic's best image; the chapter prints nothing like it.
• A persistent badge for the coprime assumption, introduced in section 4 and still visible at section 9 so the collision is seen rather than described. Standard schematic.
• Two number strips, squares and doubled squares, that can be slid past each other without ever aligning. Standard schematic, not in the book.
• No textbook figure is required. §1.3 carries no artwork on pp. 6–9.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, printed Chapter 1 "Real Numbers", §1.3 "Revisiting Irrational Numbers": the definition restated and the list of familiar irrational numbers on p. 6; Theorem 1.3 stated at the foot of p. 6, where its proof also opens with the assumption made for contradiction, and running over onto p. 7.
• Theorem 1.2 on p. 6, which the proof calls twice.
• Deliberate cross-reference outside this chapter's pages: §1.3, p. 6, names Appendix 1 as the place where proof by contradiction is treated at greater length. That appendix, "Proofs in Mathematics", is printed on pp. 218–238 of the same book, well outside this chapter's own range of pp. 1–9.
• §1.4 "Summary", p. 9, third listed point.